Characterizing maximally many-body entangled fermionic states by using $M$-body density matrix

TL;DR

This paper investigates the entanglement structure of fermionic many-body states using $M$-body reduced density matrices, establishing bounds, connections to hypergraph structures, and analyzing entanglement in random states relevant for quantum computing.

Contribution

It introduces a framework linking $M$-body entanglement in fermionic states to hypergraph structures and characterizes maximally entangled states via $t$-designs and random matrix theory.

Findings

Maximal entanglement corresponds to $t$-design hypergraphs.

Random fermionic states become maximally entangled in large dimensions.

Upper bounds on $M$-body entanglement follow a volume law.

Abstract

Fermionic Hamiltonians play a critical role in quantum chemistry, one of the most promising use cases for near-term quantum computers. However, since encoding nonlocal fermionic statistics using conventional qubits results in significant computational overhead, fermionic quantum hardware, such as fermion atom arrays, were proposed as a more efficient platform. In this context, we here study the many-body entanglement structure of fermionic $N$-particle states by concentrating on $M$-body reduced density matrices (DMs) across various bipartitions in Fock space. The von Neumann entropy of the reduced DM is a basis independent entanglement measure which generalizes the traditional quantum chemistry concept of the one-particle DM entanglement, which characterizes how a single fermion is entangled with the rest. We carefully examine upper bounds on the $M$-body entanglement, which are analogous to the volume law of conventional entanglement measures. To this end we establish a connection between $M$-body reduced DM and the mathematical structure of hypergraphs. Specifically, we show that a special class of hypergraphs, known as $t$-designs, corresponds to maximally entangled fermionic states. Finally, we explore fermionic many-body entanglement in random states. We semianalytically demonstrate that the distribution of reduced DMs associated with random fermionic states corresponds to the trace-fixed Wishart-Laguerre random matrix ensemble. In the limit of large single-particle dimension $D$ and a non-zero filling fraction, random states asymptotically become absolutely maximally entangled.